Isometric embeddings of subdivided wheels in hypercubes

The k-wheel Wk is the graph obtained as a join of a vertex and the cycle of length k. It is proved that a subdivided wheel G embeds isometrically into a hypercube if and only if G is the subdivision graph S(K4) of K4 or G is obtained from the wheel Wk (k⩾3) by subdividing any of its outer-edges with an odd number of vertices.